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11 Jul 2025, 21:26
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"At a company with 36 employees, 23 are assigned to Project Alpha, and 18 are assigned to Project Beta. How many employees are assigned to both projects?"
From stem:
(The exclamation mark means "not", so !Alpha means Not Alpha)
Then we can infer !Alpha and !Beta:
We are asked to answer (Beta & Alpha).
(1) Five employees are not assigned to either project.
With (!Alpha & !Beta) and our inferred numbers from stem we can calculate all values, as shown in the table below.
It is sufficient? Yes, it is. Eliminate answer choices B, C, and E.
(2) Exactly 21 employees are assigned to only one of the two projects.
x + y = 21
At a first glance it looks not sufficient. But, analyzing carefully, we can test values:
x must be lower or equal than min(Beta, !Alpha) = min(18, 13) = 13
y must be lower or equal than min(Alpha, !Beta) = min(18, 23) = 18
Let's analyze if it's possible to arrive to only one value. As x has the "lowest maximum" I'll use it as baseline and define y as 21 - x.
if x = 13 then y = 8 -> not possible (v inconsistent 0 AND 10)
if x = 12 then y = 9 -> not possible (v inconsistent 1 AND 9)
if x = 11 then y = 10 -> not possible (v inconsistent 2 AND 8)
if x = 10 then y = 11 -> not possible (v inconsistent 3 AND 7)
if x = 09 then y = 12 -> not possible (v inconsistent 4 AND 6)
if x = 08 then y = 13 -> possible (v consistent 5 AND 5)
if x = 07 then y = 14 -> not possible (v inconsistent 6 AND 4)
if x = 06 then y = 15 -> not possible (v inconsistent 7 AND 3)
if x = 05 then y = 16 -> not possible (v inconsistent 8 AND 2)
if x = 04 then y = 17 -> not possible (v inconsistent 9 AND 1)
if x = 03 then y = 18 -> not possible (v inconsistent 10 AND 0)
So, we can arrive in a unique solution.
It is sufficient? Yes, it is. Eliminate answer choice A.
Answer = D Staments (1) and (2) alone are sufficient.
From stem:
(The exclamation mark means "not", so !Alpha means Not Alpha)
| Alpha | !Alpha | Total | |
| Beta | 18 | ||
| !Beta | |||
| Total | 23 | 36 |
Then we can infer !Alpha and !Beta:
| Alpha | !Alpha | Total | |
| Beta | ? | 18 | |
| !Beta | 18 | ||
| Total | 23 | 13 | 36 |
We are asked to answer (Beta & Alpha).
(1) Five employees are not assigned to either project.
With (!Alpha & !Beta) and our inferred numbers from stem we can calculate all values, as shown in the table below.
| Alpha | !Alpha | Total | |
| Beta | ? = 10 | 8 | 18 |
| !Beta | 13 | 5 | [color=#2ecc40]18[/color] |
| Total | 23 | [color=#2ecc40]13[/color] | 36 |
It is sufficient? Yes, it is. Eliminate answer choices B, C, and E.
(2) Exactly 21 employees are assigned to only one of the two projects.
| Alpha | !Alpha | Total | |
| Beta | w | x | 18 |
| !Beta | y | v | [color=#2ecc40]18[/color] |
| Total | 23 | [color=#2ecc40]13[/color] | 36 |
x + y = 21
At a first glance it looks not sufficient. But, analyzing carefully, we can test values:
x must be lower or equal than min(Beta, !Alpha) = min(18, 13) = 13
y must be lower or equal than min(Alpha, !Beta) = min(18, 23) = 18
Let's analyze if it's possible to arrive to only one value. As x has the "lowest maximum" I'll use it as baseline and define y as 21 - x.
if x = 13 then y = 8 -> not possible (v inconsistent 0 AND 10)
if x = 12 then y = 9 -> not possible (v inconsistent 1 AND 9)
if x = 11 then y = 10 -> not possible (v inconsistent 2 AND 8)
if x = 10 then y = 11 -> not possible (v inconsistent 3 AND 7)
if x = 09 then y = 12 -> not possible (v inconsistent 4 AND 6)
if x = 08 then y = 13 -> possible (v consistent 5 AND 5)
if x = 07 then y = 14 -> not possible (v inconsistent 6 AND 4)
if x = 06 then y = 15 -> not possible (v inconsistent 7 AND 3)
if x = 05 then y = 16 -> not possible (v inconsistent 8 AND 2)
if x = 04 then y = 17 -> not possible (v inconsistent 9 AND 1)
if x = 03 then y = 18 -> not possible (v inconsistent 10 AND 0)
So, we can arrive in a unique solution.
It is sufficient? Yes, it is. Eliminate answer choice A.
Answer = D Staments (1) and (2) alone are sufficient.
11 Jul 2025, 21:39
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Bunuel
Total =36
Alpha (A) = 23
Beta (B) = 18
Both=?
Statement 1:
Neither (N)= 5
Total -N = A+B -Both
36-5= 23+18-Both
=> Both =10
Sufficient
Statement 2:
Exactly 21 assigned to only one of the two projects
A-Both + B-Both = 21
=> 23-Both + 18-Both =21
=> Both =10
Sufficient
ANSWER: D
11 Jul 2025, 22:13
Kudos
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We have A =23, B= 18 => find Both= ?
(1) Total = A+B-Both+Neither => 36=23+18- Both +5 => Both =10
(2) a is the number of employees assigned only to Project A, b is the number of employees assigned only to Project B
We have a+b=21=> A-Both +B-Both= A+B -2Both=21=> Both=10
(1) Total = A+B-Both+Neither => 36=23+18- Both +5 => Both =10
(2) a is the number of employees assigned only to Project A, b is the number of employees assigned only to Project B
We have a+b=21=> A-Both +B-Both= A+B -2Both=21=> Both=10
Bunuel
